Thursday, August 20, 2015

August 2015 Geometry Regents -- NOT Common Core: Part 1, Multiple-Choice

Here are the questions, with answers and explanations, for the New York State Geometry Regents exam. This is the old exam, which is being retired not the Common Core exam, which can be found here.

There were 28 questions, each worth 2 credits. No partial credit. No work needed to be shown (but it would still be a good idea to work out the answers, even if no one will see it). Twenty correct answers on this part -- which is roughly two-thirds of the questions -- results in 40 credits, which curves to a grade of 65 (which is roughly two-thirds of the points). This is slightly lower than previous exams, but is a higher threshold than the Common Core exam.

Images will be added when I have the resources available. Sorry, but there are a lot of images in this exam!

Part 1

1. In triangle ABC shown below with ADC, AEB, CFE, and BFD, triangle ACE = triangle ABD. (image omitted)
Which statement must be true?

(4) <AEF = <ADF. Corresponding parts of congruent triangles are congruent.

2. In a circle whose equation is (x - 1)2 + (y + 3)2 = 9, the coordinates of the center and length of its radius are

(3) (1, -3) and r = 3. (x - h)2 + (y - k)2 = r2. h = 1, k = -3, r = 3.

3. Parallel secants FH and GJ intersect circle O, as shown in the diagram below. (image omitted)
If m(arc)FH = 106 and m(arc)GJ = 24, then m(arc)FG equals

(2) 115. Arcs FG and HJ are congruent, and FG + HJ + 106 + 24 = 360. FG + HJ = 230, FG = 115.

4. What are the coordinates of P', the image of point P(x, y) after translation T4,4?

(2) (x + 4, y + 4). Definition of a translation.

5. The statement "x > 5 or x < 3" is false when x is equal to

(4) 4. For an "OR" statement (disjunction) to be false, both parts have to be false. Four is neither greater than 5 nor less than 3.

6. Triangle JTM is shown on the graph below. (Image omitted)
Which transformation would result in an image that is not congruent to triangle JTM?

(4) D2. Dilations preserve the shape and orientation but not the size of the original shape. The image is similar but not congruent.

7. In the diagram below of triangle ABC, with CDEA and BGFA, EF || DG || CB. (image omitted)
Which statement is false?

(3) AE/AD = EC/AC. The three triangles are similar, so their corresponding sides are proportional. In Choice (3) the proportion is not set up correctly. The sides do not correspond to each other.

8. Which pair of edges is not coplanar in the cube shown below? (image omitted)

(1) EH and CD. EH sort of goes front to back while CD sort of goes left to right. They aren't in the same plane. They couldn't be lines, for example, on a single sheet of paper.

9. What is an equation of the line that passes through the point (-2,1) and is parallel to the line whose equation is 4x - 2y = 8?

(3) y = 2x + 5. Parallel means the same slope. Converting the original equation

4x - 2y = 8
- 2y = -4x + 8
y = 2x - 4
The slope of the line is 2, so eliminate choices (1) and (2). Plug in (-2, 1) into the other choices, and choice (3) is correct.

10. In triangle JKL, JL = KL. If m<J = 58, then m<L is

(2) 64. Triangle JKL is an isosceles triangle. Angle L is the vertex angle, angles J and K are the base angles, which are congruent to each other. So 58 + 58 + L = 180. L = 64.

11. The corresponding medians of two similar triangles are 8 and 20. If the perimeter of the larger triangle is 45, what is the perimeter of the smaller triangle?

(2) 18. The scale factor is 8/20, or 2/5. Multiply (45)(2/5) = 18. Note: Perimeter is one-dimensional so the change is proportional. Had it been Area, which is two dimensions, you would have had to multiply by the square of the scale factor.

12. Which construction of parallel lines is justified by the theorem "If two lines are cut by a transversal to form congruent alternate interior angles, then the lines are parallel"?

(3). Choice (3) is the only one where the alternate interior angles are marked off with a construction.

13. Given: "If a polygon is a triangle, then the sum of its interior angles is 180°."
What is the contrapositive of this statement?

(4). "If the sum of the interior angles of a polygon is not 180°, then it is not a triangle." Definition of contrapositive: IF A THEN B. ==> IF NOT B THEN NOT A.
Choice (2) is a biconditional (if and only if). Choice (3) is the inverse. Choice (4) is the converse. Note that all four statements have the same truth value.

14. In the diagram below, point P is not on line L. (image omitted)
How many distinct planes that contain point P are also perpendicular to line L?

(1) 1. Given a line and point not on that line, there is only one line parallel to the first one that goes through the point.

15. The image of triangle ABC after the transformation ry-axis is triangle A'B'C'. Which property is not preserved?

(2) orientation. A reflection will change the direction of an object. The object and its image will be congruent to each other.

16. The equations y = 2x + 3 and y = -x2 - x + 1 are graphed on the Use this space for same set of axes. The coordinates of a point in the solution of this system of equations are

(4) (-2, -1). You can plug the points into both equations and see which one is a solution to both equations. Choice (1) is right out: 2(0) + 3 =/= 1. Or you can solve the quadratic equation:

-x2 - x + 1 = 2x + 3
-x2 - 3x - 2 = 0
x2 + 3x + 2 = 0
(x + 2)(x + 1) = 0
x = - 2 or x = -1

Substitute y = 2(-1) + 3 = -2 + 3 = 1, (-1, 1)
y = 2(-2) + 3 = -4 + 3 = -1, (-2, -1)
NOTE: The fact that one x value was -1 and one y value was -1 IS A CO-INCIDENCE.

17. Which quadrilateral has diagonals that are always perpendicular bisectors of each other?

(1) square. That is a property of rhombuses, and a square is a rhombus. If it were true for (2) or (4), it would also be true for (1), so those should be eliminated immediately.

18. As shown in the diagram below, AB is a diameter of circle 0, and chord AC is drawn. (image omitted)
If m<BAC = 70, then m(arc)AC is

(1) 40. Arc AB is a semicircle, which is 180 degrees. Arc BC is twice as big as the inscribed angle, which is 2 * 70 = 140 degrees. Subtract 180 - 140 = 40 degrees.

19. In parallelogram JKLM, m<L exceeds m<M by 30 degrees. What is the measure of <J?

(2) 105o. Angle J is congruent to angle L, which it is across from. Angles L and M are supplementary because they are consecutive angles (and same-side interior angles of a transversal across two parallel lines). L + M = 180 and L = M + 30, which means L - 30 = M. So L + L - 30 = 180. 2L = 210. L = 105.

20. Which equation represents the circle shown in the graph below?

(2) (x + 5)2 + (y - 3)2 = 1. The equation of a circle, again. It has its center at (-5, 3) and a radius of 1. Flip the signs on x and y, and square r.

21. What is the measure of each interior angle in a regular octagon?

(2) 135o. First of all, you should have seen problems like this so often that you should just know the answers for equilateral triangle, square, pentagon, hexagon, octagon and decagon. (Yes, I skipped a couple, because you don't see them as much.)
(n-2) * 180 gives you the total number of degrees: 6 * 180 = 1080 degrees. Divide that by the number of angles, which is 8: 1080 / 8 = 135 degrees.

22. Points A and B are on line L, and line L is parallel to line m, as shown in the diagram below. (image omitted)
How many points are in the same plane as L and m and equidistant from L and m, and also equidistant from A and B?

(1) 1. Equidistant from L and m is a parallel line between them, running left to right. Equidistant from A and B is a line perpendicular to line L that intersects it at the midpoint of AB. The vertical line and the horizontal line intersect at exactly one point. (Two lines with different slopes in the same plane will always intersect at exactly one point.)

23. A carpenter made a storage container in the shape of a rectangular prism. It is 5 feet high and has a volume of 720 cubic feet. He wants to make a second container with the same height and volume as the first one, but in the shape of a triangular prism. What will be the number of square feet in the area of the base of the new container?

(3) 144. I read this one twice, looking for the trick. There is none. It is total misdirection. The shape of the Base does not matter. If the height is the same and the Volume is the same then the Area of the Base has to remain the same. V = Bh, so B = V/h = 720/5 = 144.

24. In triangle ABC, m<B < mltA < m<C. Which statement is false?

(1) AC > BC. The smallest angle is opposite the smallest side. The largest side is opposite the largest angle. AC < BC < AB.

25. In the diagram below of circle O with radius OA, tangent CA and secant COB are drawn. (image omitted)
If AC = 20 cm and OA = 7 cm, what is the length of OC, to the nearest centimeter?

(3) 21. Very straightforward Pythagorean Theorem problem. No tricks, no hunting for information. The tangent forms a right angle with the radius. That makes OC the hypotenuse of a right triangle. 72 + 202 = OC2. You should know that 7-20-21 is a Pythagorean Triple.
The answer is "obviously" 21 because OC is the hypotenuse, so 19 and 20 are out, and 20-7-27 is not a valid triangle, let alone a right triangle.

26. In the diagram below of triangle ABC, point H is the intersection of the three medians. (image omitted)
If DH measures 2.4 centimeters, what is the length, in centimeters, of AD?

(3) 7.2. Three medians meet at a centroid. The centroid divides median DA into the following relationships: DH is half the size of HA. DH is one-third the size of DA. And HA is two-thirds the size of DA. Multiply 2.4 * 3 = 7.2.

27. Which set of numbers could be the lengths of the sides of an isosceles triangle?

(2) {3, 3, 5}. First, eliminate (3) because it isn't even isosceles. Choices (1) and (4) do not form triangles because the sum of the smaller sides is not greater than the longest side.

28. In the diagram below of right triangle ABC, CD is the altitude to hypotenuse AB, AD = 3, and DB = 4. (image omitted)
What is the length of CB?

(3) 2*Sqrt(7). There are three right triangles in the picture. The large one, which is divided into two smaller ones. They are all similar, so their corresponding sides are proportional. Therefore, you can write 4/x = x/(3+4). So x2 = 4*7 = 28, and x = sqrt(28) = sqrt(4)*sqrt(7) = 2*sqrt(7).

That's the end of Part 1. Part 2 will hopefully be uploaded soon, and there will be a link here.


Unknown said...

I'm tutoring a student who failed the June 2015 and August 2015 Geometry Regents. Your explanations for arriving at the answers to Part I questions were very helpful to me and will make it much easier for me to show my student how to write out the work.

(x, why?) said...

Thanks for the kind words.

I'm working my way through three exams as time permits.

Anonymous said...

7-20-21 is not a triple.

49 + 400 = 449

21^2 = 441

(x, why?) said...

Good catch. I don't know where my mind was. The triple is 7-24-25. And 20-21-29 is a different triple.

That said, the answer is still "obviously" 21 because it is the hypotenuse, so 19 and 20 are out, and 20-7-27 is not a valid triangle, let alone a right triangle.

Answer modified above.